Let $B$ be a fractional Brownian motion with Hurst index $H in (0,1)$. Denote by $x_1 < x_2 < cdots$ the positive, real zeros of the Bessel function $J_{-H$ of the first kind of order $-H$, and let $y_1 < y_2 < cdots$ be the positive zeros of $J_{1-H$. We prove the series representation begin{equation* B_t = sum_{n=1^infty frac{sin x_n t{x_n, X_n + sum_{n=1^infty frac{1-cos y_n t{y_n, Y_n, end{equation* where $X_1, X_2, ldots$ and $Y_1, Y_2, ldots$ are independent, Gaussian random variables with mean zero and $Var X_n = 2c_H^2x_n^{-2HJ^{-2_{1-H(x_n)$, $Var Y_n = 2c_H^2y_n^{-2HJ^{-2_{-H(y_n)$, where the constant $c_H^2$ is defined by $c_H^2 = pi^{-1Gamma(1+2H)sin pi H$. With probability $1$, both random series converge absolutely and uniformly in $t in [0,1]$.

, ,
,
CWI
CWI. Probability, Networks and Algorithms [PNA]
Stochastics

Dzhaparidze, K.O, & van Zanten, J.H. (2002). A series expansion of fractional Brownian motion. CWI. Probability, Networks and Algorithms [PNA]. CWI.